WoW that Is Cool ! You are very lucky !
kinda looks like a tree Peony in the mix,
I hope someone can help You with more info.
Olgaflowers

The double blooms do seem to come from tp foliage.
Seems a little odd that a herb and a tp would bloom at same time.
Maybe your plant was originally a tp grafted onto herb root.

I have linked to information on your tree peony. Put a marker on these stems and any others that connect to them for now. This fall when the foliage dies down very carefully remove the foliage that you have not marked. If you can dig the tree peony part without digging the rest try to do so. The regular peony grows faster than the tree peony and if they are still connected may take nutriants from the tp. With luck and great care you may be able to separate the plants.

I would recomend that you use a butter knife or some rounded tool to remove the soil around the tp part to see if it has it's own roots. You may have to dig part of the regular peony to separate them. If the tp has it's own roots you may be able to just remove the part with the roots. You may need a small saw to separate. To have a larger tp in the future try to bury most of the stems under ground. Roots and future flowering stems will come from the buried portion.

Here is a link that might be useful: link to your tree peony

maifleur, I think this may be the link you were looking for.

Here is a link that might be useful: Paeonia 'Souvenir du Professor Maxime Cornu'

Well, I would be mighty surprised and pleased too. Good luck at getting them separated.

Sue

Thank you for your help, this has been one of the more interesting developments in my garden this year. I have several herbaceous peonies, but this is the first tree peony in my garden.

I had wondered about the different leaves, and hate to admit, even considered cutting them off, I'm so glad I didn't. More experience with peonies would be a good thing right now, but will be ever so cautious when looking into splitting them, I would hate to lose either.